BALKAN-SL 2008 G2

Given a scalene acute triangle $ABC$ with $AC>BC$ let $F$ be the foot of the altitude from $C$. Let $P$ be a point on $AB$, different from $A$ so that $AF=PF$. Let $H,O,M$ be the orthocenter, circumcenter and midpoint of $[AC]$. Let $X$ be the intersection point of $BC$ and $HP$. Let $Y$ be the intersection point of $OM$ and $FX$ and let $OF$ intersect $AC$ at $Z$. Prove that $F,M,Y,Z$ are concyclic.